By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Inverses By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: November 17, 2005

Definition Inverse Relation  A relation obtained by switching the coordinates of each ordered pair. Jeff Bivin -- LZHS

INVERSE RELATIONS x y { (3, 8) } relation Domain Range 3 8 inverse { (8, 3) } Jeff Bivin -- LZHS

y = x Relation  { (1, 4), (4, 6), (-3, 2), (-4, -2), (-1,5), (0, 1) } Inverse  { (4, 1), (6, 4), (2, -3), (-2, -4), (5, -1), (1, 0) } y = x Jeff Bivin -- LZHS

y = x Relation  {(-4,-6), (1,4), (2, 6), (-1,0), (-4,3), (4,-2)} Inverse  {(-6,-4), (4,1), (6, 2), (0,-1), (3,-4), (-2,4)} y = x Jeff Bivin -- LZHS

f(x)= x2 y = x Jeff Bivin -- LZHS

f(x)= x2 y = x Jeff Bivin -- LZHS

G(x) y = x Jeff Bivin -- LZHS

G(x) y = x Jeff Bivin -- LZHS

G(x) y = x Jeff Bivin -- LZHS

G(x) y = x Jeff Bivin -- LZHS

f(x)= x3 y = x Jeff Bivin -- LZHS

Find the inverse Is this a function? YES Jeff Bivin -- LZHS

Find the inverse Is this a function? NO Jeff Bivin -- LZHS

Find the inverse Is this a function? NO Jeff Bivin -- LZHS

f(g(x)) = x and g(f(x)) = x Inverse functions Two functions, f(x) and g(x), are inverses of each other if and only if: f(g(x)) = x and g(f(x)) = x Jeff Bivin -- LZHS

Are these functions inverses? Therefore: Inverses Jeff Bivin -- LZHS

Are these functions inverses? Therefore: NOT Inverses Jeff Bivin -- LZHS

One-to-One functions A function is one-to-one if no two elements in the domain of the function correspond to the same element in the range. Domain Range 2 1 F(x) One-to-One 5 -5 9 4 Jeff Bivin -- LZHS

f(x)= x2 y = x Jeff Bivin -- LZHS